On the Syntactic Complexity of Darwinian Membrane Systems

Membrane or P systems form a distributed parallel model of computing which is obtained as an abstraction from the structure and functioning of living cells. In this paper we consider a very basic membrane system and add to it checkers which are special finite automata to check whether or not a configuration of the membrane system is "good" or "bad". The computation is only continued if the configuration is good, otherwise it stops without a result. Such membrane systems are called Darwinian P systems. There are three parameters characterizing the size of a system, the number of membranes, the number of checkers, and the size of the checkers measured by the number of states. We prove that we can generate all sets of Parikh vectors of recursively enumerable languages if we restrict the number and size of checkers to one checker with six states or to five checkers with two states. Moreover, we prove that Darwinian systems with one membrane and checkers with at most two states are also sufficient to generate all Parikh sets of recursively enumerable languages. The latter result is optimal since Darwinian systems with checkers with only one state generate only sets of Parikh vectors of ET0L languages. All results are valid for length sets instead of sets of Parikh vectors, too.